Symmetry and conservation laws · Quantum Mechanics

Intuition

Put the last three lessons together and a theorem appears. If a Hamiltonian is unchanged by a continuous family of transformations, it commutes with their generator, and a quantity that commutes with the Hamiltonian is conserved — not only its average, but the probability of each of its values. Time translations conserve energy, space translations momentum, rotations angular momentum, and a change of every charged particle’s phase conserves charge. In quantum mechanics the argument also runs backwards: every conserved quantity generates a symmetry.

A lake with no current and no wind looks the same everywhere, and a boat drifting on it keeps its speed and direction. Remove the sameness — a current here, a breeze there — and the boat’s motion changes. Conservation is sameness seen from the inside.

Symmetry and conservation

Noether’s theorem in quantum mechanics: invariance of $\hat{H}$ under $e^{-is\hat{G}/\hbar}$ for all $s$ is the same as $[\hat{G},\hat{H}]=0$ , which is the same as $\hat{G}$ being conserved.

\begin{aligned}&e^{-is\hat{G}/\hbar}\,\hat{H}\,e^{is\hat{G}/\hbar}=\hat{H}\ \ \text{for all}\ s\\ &\iff[\hat{G},\hat{H}]=0\\ &\iff\frac{d}{dt}\langle\hat{G}\rangle=0\ \ \text{in every state}\end{aligned}

The table

Translations in time: energy. Translations in space: momentum. Rotations: angular momentum. A common change of phase of every charged particle: charge.
A conserved quantity keeps its whole distribution: the probability of each of its eigenvalues is constant, since $e^{-i\hat{H}t/\hbar}$ maps each eigenspace of $\hat{G}$ to itself.

Invariance implies conservation

e^{-is\hat{G}/\hbar}\hat{H}e^{is\hat{G}/\hbar}=\hat{H}\ \ \forall s\ \Longrightarrow\ \frac{d}{dt}\langle\hat{G}\rangle=0

The transformed Hamiltonian does not depend on $s$ , so its derivative at $s=0$ vanishes; that derivative is a multiple of the commutator of $\hat{G}$ with $\hat{H}$ . A vanishing commutator makes the rate of change of every average of zero.

Proof steps

\hat{H}(s)=e^{-is\hat{G}/\hbar}\,\hat{H}\,e^{is\hat{G}/\hbar}=\hat{H}\ \ \text{for all}\ s

The Hamiltonian is unchanged by every member of the family.

\frac{d\hat{H}(s)}{ds}\Big|_{s=0}=-\frac{i}{\hbar}\bigl(\hat{G}\hat{H}-\hat{H}\hat{G}\bigr)

Differentiate each exponential at $s=0$ .

[\hat{G},\hat{H}]=0

The derivative of something constant is zero.

\frac{d}{dt}\langle\hat{G}\rangle=\frac{i}{\hbar}\bigl\langle[\hat{H},\hat{G}]\bigr\rangle=0

The rate of change of any average, from the lesson on conserved quantities.

Applications

Practice

Invariance Means Conservation

If a Hamiltonian is unchanged by a continuous family of transformations, it commutes with their generator, and the generator’s whole distribution is constant in time.

[\hat{G},\hat{H}]=0\ \Longrightarrow\ \frac{d}{dt}\langle\hat{G}\rangle=0

Try it

A Hamiltonian is unchanged by every rotation about the $z$ axis. What is conserved?